Thank you for your quite thorough response to my query. One quick
question.
Stata will allow me to combine your solutions 1 and 3. And it does (at
least things move in the directions I would expect) what I would expect
(narrower confidence intervals for the estimates derived using
solution 3.
But something about it doesn't seem valid. Any thoughts?

Solution 1 in Bill Gould's post refers to fitting a Cox model where you
cluster on groups (centers in this example). Solution 3 refers to
instead
stratifying on these centers. As such, combining 1 and 3 would both
cluster
and stratify on center, i.e.

. stcox x1 x2, strata(center) vce(cluster center)

Daniel expresses concern that the above may not be valid. It is
perfectly
valid. As Bill explained, stratifying on center allows for different
baseline
hazard functions for each center. However, the effects of the
covariates x1
and x2 are assumed to be proportional on the hazard and, more
importantly,
shared across centers. In estimating the standard errors of these
(overall) covariate effects, one could also choose to cluster on center.

You stratify on center because you want the flexibility afforded by
having
uniquely shaped hazards for each group. However, once you stratify,
you are
still faced with deciding whether the subjects within each center are
to be treated as independent given the shared baseline hazard. If
you do not
wish to make this additional assumption, then you should cluster on
center as
well.